15 Non-local massive gravity

The ghost-free theory of massive gravity proposed in Part II as well as the Lorentz-violating theories of
Section 14 require an auxiliary metric. New massive gravity, on the other hand, can be formulated in a way
that requires no mention of an auxiliary metric. Note however that all of these theories do break one copy of
diffeomorphism invariance, and this occurs in bi-gravity as well and in the zwei dreibein extension of new
massive gravity.

One of the motivations of non-local theories of massive gravity is to formulate the theory without any reference
metric.34
This is the main idea behind the non-local theory of massive gravity introduced
in [328*].35

Starting with the linearized equation about flat space-time of the Fierz–Pauli theory

where is the linearized Einstein tensor, this modified Einstein equation can be
‘covariantized’ so as to be valid about for any background metric. The linearized Einstein tensor
gets immediately covariantized to the full Einstein tensor . The mass term, on
the other hand, is more subtle and involves non-local operators. Its covariantization can take
different forms, and the ones considered in the literature that do not involve a reference metric are

where is the covariant d’Alembertian and represents the retarded
propagator. One could also consider a linear combination of both possibilities. Furthermore, any of
these terms could also be implemented by additional terms that vanish on flat space, but one
should take great care in ensuring that they do not propagate additional degrees of freedom (and
ghosts).

Following [328*] we use the notation where designates the transverse part of a tensor. For any tensor
,

The theory propagates what looks like a ghost-like instability irrespectively of the exact formulation
chosen in (15.2*). However, it was recently argued that the would-be ghost is not a radiative degree of
freedom and therefore does not lead to any vacuum decay. It remains an open question of whether the
would be ghost can be avoided in the full nonlinear theory.

The cosmology of this model was studied in [395, 228]. The new contribution (15.2*) in the Einstein
equation can play the role of dark energy. Taking the second formulation of (15.2*) and setting the
graviton mass to , where is the Hubble parameter today, reproduces the
observed amount of dark energy. The mass term acts as a dark fluid with effective time-dependent
equation of state , where is the scale factor, and is thus
phantom-like.

Since this theory is formulated at the level of the equations of motion and not at the level of the action
and since it includes non-local operators it ought to be thought as an effective classical theory. These
equations of motion should not be used to get some insight on the quantum nature of the theory nor on its
quantum stability. New physics would kick in when quantum corrections ought to be taken into account. It
remains an open question at the moment of how to embed nonlocal massive gravity into a consistent
quantum effective field theory.

Notice, however, that an action principle was proposed in Ref. [402*], (focusing on four dimensions),

where the function is defined as

where is the Euler’s constant, is the incomplete gamma function,
is a integer and is a real polynomial of rank . Upon deriving the equations
of motion we recover the non-local massive gravity Einstein equation presented above [402],

up to order corrections. We point out however that in this action derivation principle the operator
likely correspond to a symmetrized Green’s function, while in (15.2*) causality requires to
represent the retarded one.

We stress, however, that this theory should be considered as a classical theory uniquely and not be
quantized. It is an interesting question of whether or not the ghost reappears when considering
quantum fluctuations like the ones that seed any cosmological perturbations. We emphasize
for instance that when dealing with any cosmological perturbations, these perturbations have
a quantum origin and it is important to rely of a theory that can be quantized to describe
them.